There's some value in that. Not as a long term principle, but I am all for having students learn a variety of methods and have to spend some time working through those methods. But only to make sure the students are giving the different methods a shot and actually finding the one that makes the most sense to them, rather than just sticking with whatever was learned first. If they're never given the chance to just work with their method of choice after, it's pointless.

This is a great point. I found myself both admiring and hating some of my son's school work recently in that it illustrated and described mental problem solving exceptionally well, then gruelingly pushed him to repeatedly explain the model rather than simply use it. Pages upon pages of dissection and regurgitation. My son could understand the model just fine, but was perplexed as to why he had to go over it like that.

That's because public school uses memorization as a cheap stand-in for understanding. It's easy to internalize 2+2=4, but it's impossible to internalize the quadratic equation. At some point, you need to just know the procedure and it's application, and most importantly, recognize when to use it in novel situations.